Hi, and welcome. My name is Anthony Varela, and today, we're going to be talking about slope in a couple of different contexts. And specifically, we're going to be talking about slope as an average rate of change. So we're going to look at average rates of change on graphs. And then we're going to wrap up by looking at positive, negative, and zero rates of change and interpreting them within a couple of different contexts.
So to think about average rate of change, let's imagine a flight going from Minneapolis, Minnesota to Seattle, Washington. So I looked up the distance in the air between these two cities, and it's about 1,400 miles. And I also looked up direct flights from Minneapolis to Washington and checked out the flight time. And it seemed pretty typical that three and a half hours would be the flight time between these two cities.
So we can use this distance and this time to calculate the speed of the aircraft. So 1,400 miles divided by 3.5 hours would be 400 miles per hour. Now this 400 miles per hour represents the average speed of the plane, so the plane doesn't instantaneously jump to 400 miles per hour as soon as it takes off. And as it approaches Seattle, it doesn't suddenly stop, go from 400 to zero.
There's obviously some build up and then some slow down as it gets close to its destination. So here is a graph of what this trip could look like, so we have time on the x-axis. And we have distance on the y-axis. I mean, you can see at the origin, this is zero time and zero miles traveled.
And then when we go three and a half hours, we've traveled 1,400 miles. And we notice, it's not a straight line, because the airplane could be going faster or slower during different parts of the flight. But after three and a half hours, we're going to reach 1,400 miles.
So taking a look at connecting the starting point and our ending point, this makes a straight line. Because we've just took two points and connected them. And if we look at the slope of this line, so the rise over the run, we could see that the rise is 1,400 miles and the run is 3.5 hours. And that's our average speed of 400 miles per hour.
So when we're talking about average rate of change, we can also talk about slope. So let's take a look at another example. This picture here shows a sea level and then the sea floor getting deeper and deeper as we walked out into the ocean. So if I'm sitting here at the beach, and I walk out then 15 feet, that's the point where I am completely underwater. So I'd be six feet below sea level here.
So let's think about this average rate of change as I am walking out 15 feet. Well, we can see then that my rise is negative six feet and my run is 15 feet. So I can simplify this and say, negative 2/5. So if I create a straight line connecting then my starting point and a finishing point, this should have a slope of negative 2/5.
So let's go ahead and put this on a graph to confirm this. So here is our origin, zero, zero. And we can see that we have a rise of negative two and a run of five, so we're at our line. A rise of negative two and a run of five, we're at our line. A rise of negative two, a run of five, we're at that line.
So slope and average rate of change describe the same thing. That's the big idea here. So lastly, I'd like to talk about positive, negative, and zero rates of change. So we're going to be taking a look at graphs and interpreting the context.
So here's my first graph. We have time on the x-axis, and we have speed on the y-axis. And we notice that this is a positive slope. Because as we read our graph left to right, the line is going upwards. This is increasing interval, so what we can gather then from this context is that speed increases with time. So as time increases, speed increases.
And how can we describe an increasing speed? This would represent then acceleration. So if this graph here is showing motion of an object, this is accelerating. Its speed is getting faster and faster.
Let's take a look at this graph. Here, we know that we have a negative slope, because our line is approaching negative infinity as we read left to right. And we have miles here on the x-axis, and we have gallons of gas on the y-axis. So this represents then gallons decreasing with distance.
This makes sense. If you're driving your car, the more miles you drive, the fewer gallons of gas you're going to have, until you fill up again. So this graph then here represents a loss of gallons depending on how many miles you travel.
And finally, we're going to look at a graph with a zero rate of change. So this looks like a horizontal line on the graph. This represents a slope of zero. So looking at how we've labeled our axes, we have time on the x-axis. We have account balance on the y-axis, so what this is saying then is that our balance is not changing, at least in the time interval that we see here on the graph.
So this represents zero financial growth. We are not gaining interest over this time interval that we have on our graph. We're not making any deposits, or withdrawals, or purchases. Absolutely zero financial growth. The balance is not changing.
So let's review slope in context. Our big idea here was that average rate of change can represent the slope of a line, so we saw this with our travel example from Minneapolis, Minnesota to Seattle, Washington. The average speed of the plane is 400 miles per hour. We saw an example of walking out into the ocean, where the average rate of change of this incline here, or this, I should say, decline, it had a slope of negative 2/5.
We also looked at positive, negative, and zero rates of change. Our positive rate of change had to do with acceleration, so speed increasing with time. Our negative rate of change example was in the context of losing gallons of gas as your driving, and our zero rate of change, our example was an account balance that wasn't gaining interest. It wasn't losing money. It had zero financial growth. So thanks for watching this tutorial on slope in context. Hope to see you next time.
